Hydrodynamic Limit of Brownian Particles Interacting with Short and Long Range Forces

Details Hydrodynamic Limit of Brownian Particles Interacting with Short and Long Range Forces Hydrodynamic Limit of Brownian Particles Interacting with Short and Long Range Forces

1998-09-24

arxiv.org/abs/cond-mat/9809331v1

Physics

Condensed Matter

Statistical Mechanics

37 pages, in TeX (compile twice), to appear on J. Stat. Phys., e-mail addresses: pbutta@math.rutgers.edu, lebowitz@math.rutger

Scientific paper

We investigate the time evolution of a model system of interacting particles, moving in a $d$-dimensional torus. The microscopic dynamics are first order in time with velocities set equal to the negative gradient of a potential energy term $\Psi$ plus independent Brownian motions: $\Psi$ is the sum of pair potentials, $V(r)+\gamma^d J(\gamma r)$, the second term has the form of a Kac potential with inverse range $\gamma$. Using diffusive hydrodynamical scaling (spatial scale $\gamma^{-1}$, temporal scale $\gamma^{-2}$) we obtain, in the limit $\gamma\downarrow 0$, a diffusive type integro-differential equation describing the time evolution of the macroscopic density profile.

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